Pentagon (EntityTopic, 12)
From Hi.gher. Space
The pentagon can be seen as the two-dimensional analog to the dodecahedron in 3D and the cosmochoron in 4D. It is also the lowest-dimensional member of the ursatopes, with a (trivial) digonal base.
coordinates
The coordinates of a regular pentagon centered at the origi and having side length 2 are:
(√((10+2√5)/5), 0)
(√((5-√5)/10), ±φ)
√((5+2√5)/5), ±1)
Where φ is the golden ratio (1+√5)/2.
Equations
- The hypervolumes of a pentagon with side length l are given by:
total edge length = 5l
area = ^{1}∕_{4} · √(25+10√5) · l^{2}
Incidence matrix
Usage as facets
- 12× 1-facets of a dodecahedron
- 12× 1-facets of a rhodomesohedron
- prism: 2× 1-facets of a pentagonal prism
- 12× 1-facets of a truncated icosahedron
- pyramid: 1× 1-facets of a pentagonal pyramid
- 12× 1-facets of a rhodoperihedron
- 60× 1-facets of a pentagonal hexecontahedron
- 24× 1-facets of a pentagonal icositetrahedron
- 12× 1-facets of a snub dodecahedron
- 6× 1-facets of a associahedron
- 4× 1-facets of a digon-unpinched pentagonal prism
- 1× 1-facets of a pentagonal cupola
- 3× 1-facets of a tridiminished icosahedron (named pen)
- 4× 1-facets of a bilunabirotunda
- 2× 1-facets of a hexagonal pyrawedge
- 2× 1-facets of a (dual of metabidiminished icosahedron)
- 1× 1-facets of a digon-unpinched square pyramid
- 2× 1-facets of a metabidiminished icosahedron (named rroob)
- 3× 1-facets of a triangular hebesphenorotunda (named pen)
- 1× 1-facets of a parabiorthotriangulated orthopinched triangular cupola
- 1× 1-facets of a pentagonal cupolawedge
- 1× 1-facets of a pentagonal transcupolawedge
- 2× 1-facets of a square transcupolawedge (named ryopc)
- 1× 1-facets of a triangulated pinched triangular cupola (named white)
- 720× 2-facets of a cosmochoron
- 72× 2-facets of a bixylodiminished hydrochoron (named pentagons)
- 288× 2-facets of a rectified snub demitesseract
- 24× 2-facets of a castellated rhodoperihedral prism
- 60× 2-facets of a castellated rhodoperihedral prism
- 12× 2-facets of a D4.3.1 dual
- 288× 2-facets of a truncated snub demitesseract
- 96× 2-facets of a D4.11
- 4× 2-facets of a bilunabirotunda pseudopyramid
- 24× 2-facets of a castellated rhodopantohedral prism (named 5 horz)
- 120× 2-facets of a castellated rhodopantohedral prism (named 2-2 vert)
- 3× 2-facets of a D4.16
- 3× 2-facets of a triangular hebesphenorotunda pseudopyramid (named pen)
- 12× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 12× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named ggbbp)
- 12× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named rroob)
- 12× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named ggbbp)
- 24× 2-facets of a D4.7 (named rcbpp)
- 12× 2-facets of a D4.7 (named ooyyg)
- 12× 2-facets of a D4.7 (named pppps)
Notable Dishapes | |
Flat: | triangle • square • pentagon • hexagon • octagon • decagon |
Curved: | circle |